Gradient and curl optical torques

Optical forces and torques offer the route towards full degree-of-freedom manipulation of matter. Exploiting structured light has led to the discovery of gradient and curl forces, and nontrivial optomechanical manifestations, such as negative and lateral optical forces. Here, we uncover the existence of two fundamental torque components, which originate from the reactive helicity gradient and momentum curl of light, and which represent the rotational analogues to the gradient and curl forces, respectively. Based on the two components, we introduce and demonstrate the concept of lateral optical torques, which act transversely to the spin of illumination. The orbital angular momentum of vortex beams is shown to couple to the curl torque, promising a path to extreme torque enhancement or achieving negative optical torques. These results highlight the intersection between the areas of structured light, Mie-tronics and rotational optomechanics, even inspiring new paths of manipulation in acoustics and hydrodynamics.


Reviewer #2 (Remarks to the Author):
The manuscript by Xu and coworkers presents a general framework for calculating and understanding the torque exerted on a Mie particle by an arbitrary electromagnetic field.
The framework allows to calculate torques in general case, where electromagnetic field carries both SAM and OAM, and the particles are described in terms of multipolar expansion.The manuscript shows that for multipoles higher than a dipole, torques from helicity gradient and momentum-curl of light arise and discuss the scenarios in which negative optical torque (NOT) and lateral optical torque (LOT) are created (using structured light beams).
The authors claim to "theoretically uncover and predict the existence of two fundamental torque components", which I believe is exaggerated.The torque terms proportional to helicity gradient and curl of momentum are already present in e.g.Phys.Rev. A 92, 043843.
The novelty lies in the realization that these terms can (under idealized conditions) produce LOT.
The manuscript is clear and well written, and the presented theoretical torque framework is mathematically very elegant.However, I believe its relevance for experimentalists is limited, since only ideal spheres are considered.Since the LOT and NOT are only present for larger Mie particles (outside dipolar approximation), considering anisotropic particles is very important.Some anisotropy is necessary so that the rotation due to torque is detectable, and the anisotropy-related effects such as restoring torques and scattering torques will likely prevail in experiments.Therefore, including the particles anisotropy is necessary to provide a truly realistic pathway towards observing LOT in experiment.
Finally, the discussion of the torque in the optical spanner appears disjointed from the remainder of the manuscript.I would suggest to omit the "optical spanner" section, so that the manuscript is more focused.
Overall, I miss the broader relevance and the groundbreaking character that would make the manuscript also interesting to experimentalists.Such broader relevance could for instance be expanding the framework to include non-spherical particles.In addition, I am not convinced LOT is interesting/relevant.I therefore cannot recommend publication of the manuscript.It may be better suited to a more specialized journal.
More detailed remarks: 1.When describing the Eq. ( 1) the authors only speak about the incident field.I find the discussion of the interaction of the particle with its own scattered field missing.In realistic settings (e.g.Phys.Rev. Lett.129, 023602) interaction with scattered field leads to torques larger than absorption.
2. It would be interesting to include the discussion of the angular momentum conservation in the context of LOT.

Response to Reviewer #1
The reviewed paper presents a theoretical investigation of the mechanical action performed by structured light fields on small spherical particles.In contrast to the most of other works, it is concentrated on the optical torque which induces the particle spinning near its own axis.So far, only one factor was considered which induces the particle spinning: absorption of light carrying the optical angular momentum (AM); even the method for measuring small light absorption was proposed, based on the observation of light-induced particle spinning, https://doi.org/10.1364/OE.23.007152; https://doi.org/10.1364/AO.54.00F174.The authors discover additional factors able to induce the particle spinning motion in highly inhomogeneous optical fields: the reactive helicity gradient and the curl of the Poynting momentum.The actions of these factors are manifested through higher multipoles of the Lorentz -Mie expansion but, according to the presented estimations, can be observed and used in the optical-manipulation purposes.It is especially important, that these new factors are able to excite the particle spinning non-collinear with the incident-light AM (lateral optical torque) and even to act oppositely (negative optical torque).
The new data obtained in the paper disclose new features of structured light fields and new possibilities in the optical manipulations.The paper is written clearly and in the rigorous scientifically sound style (some remarks see below).I did not check the complex calculations but the results look reasonable.I think, it will be interesting and useful for a wide audience, provided that the proper corrections are made before publication.Generally, this work makes an essential new contribution in the fields of optomechanics and structured-light properties.

Response (R1.1):
We are grateful to the referee's high judgment of our manuscript.The corrections are addressed as follows.
1.The authors warn that they consider "interaction of monochromatic waves with an isotropic and neutral sphere", after which the symbols "ω" and "k" are used without explanation.Of course, these are standard notations of frequency and wavenumber but, nevertheless, certain variations in their definitions are possible; these variations (or their absence) should be clearly exposed.The necessity in formal definition is especially evident when the standard symbols are used in non-traditional way: for example, is "k" in Eq. ( 9) the same wavenumber k = ω/c as in Eq. (4), etc.? R1.2: Yes, the wave numbers in Eqs. ( 9) and (4) are the same.We explain k and ω in the revision (please see P. 2, col.1).
2. Eq. ( 9) still invokes additional questions.If u denotes "the state of polarization" and is a transverse vector (a few lines below the case "u = ex" is discussed), why does this expression omit the multiplier exp(ikzz), which is present, e.g., in (5)?Besides, the meaning of "ρ0" should be explained immediately, not after two paragraphs.R1.3: Eq. ( 9) is the expression for the field at the pupil plane z = z0, so the propagating phase term exp(ikzz0), which is a constant, can be omitted.Eq. ( 5) formulates the field in the whole 3D space, so the variable phase term exp(ikzz) cannot be dropped.We explain the meaning of "ρ0" immediately following Eq.( 9) in the revision (P.6, col.2).
3. A few lines below Eq. ( 9) one reads: "one may write the field in the focal plane, by neglecting the other two orthogonal components" --> what are the "two orthogonal components" mentioned in this fragment?R1.4:We refer to the two components orthogonal to the polarization u.For example, if u represents the left-handed circular polarization |eL, then the two orthogonal components are the longitudinal polarization and right-handed circular polarization |eR.In the revision, we rephrase the related statement for clarity (P.6, col.2). 4. P. 3, col.1: "retain the symmetry descending from the Montonen-Olive duality" --> It seems that the Montonen-Olive duality is not so well known for the community that needs no explanation; at least, a proper reference should be provided.R1.5:We thank the referee for this important comment.What the "Montonen-Olive duality" actually refers to is the dual symmetry.However, we now realize the spin torque in its form does not follow this symmetry, as it is asymmetric with respect to the exchange of the electric and magnetic vectors: E→ Bc, Bc→ −E.We ignored the difference in the prefactors of the electric and magnetic parts of the torque.For these reasons, this statement is removed in the revision.
Section "Discussion and conclusion" starts with the statement "We have built a multipole theory for the optical torque, with a classifying framework featuring two fundamental field properties" but a few lines below one reads: "First, these three aforementiond quantities…".Do the authors mean that the "aforementioned" spin is also included to their "classifying framework"?Anyway, this fragment should be presented more clearly.R1.11:We thank the referee for raising this point.Yes, the spin is also included in our classifying framework.In the revised manuscript, we rephrase the related sentence as "We have built a multipole theory for the optical torque, with a classifying framework featuring three fundamental field properties: the optical spin…".
Conclusion: the manuscript can be accepted after minor corrections.

Response to Reviewer #2
The manuscript by Xu and coworkers presents a general framework for calculating and understanding the torque exerted on a Mie particle by an arbitrary electromagnetic field.The framework allows to calculate torques in general case, where electromagnetic field carries both SAM and OAM, and the particles are described in terms of multipolar expansion.The manuscript shows that for multipoles higher than a dipole, torques from helicity gradient and momentum-curl of light arise and discuss the scenarios in which negative optical torque (NOT) and lateral optical torque (LOT) are created (using structured light beams).

Response (R2.1):
We thank the referee for his/her time and effort invested in reading and evaluating our manuscript.We also thank the referee for the constructive comments, which helped a lot to improve our manuscript.
The authors claim to "theoretically uncover and predict the existence of two fundamental torque components", which I believe is exaggerated.The torque terms proportional to helicity gradient and curl of momentum are already present in e.g.Phys.Rev. A 92, 043843.The novelty lies in the realization that these terms can (under idealized conditions) produce LOT.R2.2:We apologize for any confusion caused.Concerning the paper by one of us, Nieto-Vesperinas, PRA 92, 043843 (2015), quoted as Ref. [41] in the previous manuscript (now Ref. [40]), there are fundamental differences between this manuscript and that PRA paper.
The work of that PRA paper was restricted to dipolar particles.It is true that Nieto-Vesperinas resorted to the gradient and curl terms, in deriving the torque contributions from the conservation of spin (SAM) and of orbital angular momenta (OAM), separately.However, when one adds these SAM and OAM, thus taking fully into account the conservation of total AM, (which represents the physical observable), no such gradient and curl components appear, because they cancel out in that addition that yields the net torque on the dipolar particle, which in this dipole approximation only depends on the optical spin and the absorption cross-section of the dipolar particle, as shown in Eq. ( 1), as well as Refs.[17,18,41] quoted in the revision.
In our manuscript, multipoles of arbitrary order are addressed, and we show that the gradient and curl components are present in the net torque on multipoles higher than the dipole order, so they can be said to exist.In the revision, we remark on the results in Ref. [40] to further clarify the novelty of our paper (please see P. 3, col.2).
The manuscript is clear and well written, and the presented theoretical torque framework is mathematically very elegant.However, I believe its relevance for experimentalists is limited, since only ideal spheres are considered.Since the LOT and NOT are only present for larger Mie particles (outside dipolar approximation), considering anisotropic particles is very important.Some anisotropy is necessary so that the rotation due to torque is detectable, and the anisotropy-related effects such as restoring torques and scattering torques will likely prevail in experiments.Therefore, including the particles anisotropy is necessary to provide a truly realistic pathway towards observing LOT in experiment.R2.3: Thank you for your recognition of our theoretical framework.We also agree with you that particles in reality cannot be perfect spheres and the restoring torque could be significant for some highly anisotropic structures.In response to this referee's concern, we have added completely new section, where we perform numerical simulation on anisotropic spheres, including roughness, as well as anisotropic dimers of two spheres.As regards the scattering torque mentioned by the reviewer, we have in fact considered this contribution in our theory.In particular, our framework is derived from Supplementary Eq. ( 1), in which Tsca (l) accounts for the scattering (or recoiling) torque due to the l-order multipoles.In the following reply, we shall focus on the relevance of our theory to realistic spherical particles and nonspherical structures, by comparing it with numerical results and by the method of T matrix.
To simulate the realistic sphere-like particles, we consider spheres of varying roughness (please see particles i-iv in Fig. R1a), with the illumination of the traveling wave as an example.The optical torque is computed based on Maxwell stress tensor method, in which the electromagnetic field is obtained by finite-difference time-domain (FDTD) simulations.Figure R1b shows the calculated LOT (i.e., Tz) on the particles with varying orientation angle α at y = 0 and x = 0.35λ, where the momentum curl of illumination is maximized.The α-independent torque on the isotropic particle i is approximately −0.1 pN⋅µm, in good agreement with our analytical theory (please see red line in Fig. 2d in the manuscript).Due to the anisotropy, the torque magnitude varies with α for the rough spheres ii-iv, and larger roughness yields more significant variation.
However, the torque maintains its negative sign, irrespective to the orientation.This indicates that the rough spheres will exhibit the running state to rotate clockwise and continuously, consistent with the perfect sphere, as shown by the angular potential (Fig. R1c), which is calculated by the opposite angular integral of the torque: . It also suggests that the restoring contribution of the torque is overwhelmed by that responsible for continuous rotation.Therefore, our theory can well predict and explain the dynamic behaviors of these sphere-like structures.We then evaluate the LOT and angular potential for the dimers (i.e., particles v and vi in Fig. R1a) made of two identical Si spheres of radius r.For this highly anisotropic geometry, it is instructive to extract the nonconservative and conservative (or restoring) parts from the torque: The nonconservative α-independent part noncons The results for the smaller dimer (particle v) are shown in Fig. R1d.The total LOT, Tz, varies with the orientation angle in a sinusoidal-like way, and it changes sign with a negative derivative around α = 90° or 270°.Accordingly, an angular potential well is formed, by which the particle exhibits the locked (torsional) state (Fig. R1f).It is also noted that the washboard potential is slightly titled due to the presence of a small nonconservative component Tz ncons in Fig. R1d.However, the dynamic behavior of the larger dimer (particle vi) is quite different.As shown in Fig. R1e, Tz is always negative, because the magnitude of Tz ncons dominates over that of Tz cons .As a result, the potential well vanishes, switching the dimer to the running state (Fig. R1f).
Conventionally, the running state and washboard potential tilt are attributed to the circular polarization (or optical spin) of light [Phys.Rev. Lett. 2022, 129, 023602 (quoted as Ref. [51] in revised paper)].They are, however, explained by the momentum curl in our case, because only this quantity of illumination has the lateral component (see the subsequent paragraph for detailed proof).It also interprets the dominant restoring effect on the small dimer, because the momentum curl produces torque through higher multipole responses, which are weak for reduced particle size.
We now present a rigorous argument proving that the nonconservative component Tz ncons in Fig. R1, which causes the running state and washboard potential tilt, is traced to the momentum curl.For a dipolar dimer or other rotationally symmetrical structures (e.g., cylinders and spheroids), the torque can be derived by replacing the scalar polarizability, i.e., ( 1)   e(m) l γ = in Supplementary Eq. ( 3), by a tensor, as previous work demonstrated [ Phys. Rev. Lett. 2022, 129, 023602].However, such a treatment will be invalid for anisotropic higher multipoles, because it will break the symmetric and traceless properties of multipole moments, which are required by definition [J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962)].To address the anisotropy, we employ a generic expression of the torque [JOSA A, 2007, 24(2): 430-443]: where pml and qml are the expansion coefficients of the scattered field on the basis of vector spherical wave functions (VSWFs).They are linked to the expansion coefficients of the incident field, aml and bml, via the T matrix: ( ) ( ) Note that in Eq. (R3) the terms related to * * Re( )  z mlm'l' m'l' ml mlm'l' m'l' ml mlm'l' m'l' ml mlm'l' m'l' ml l m l l' m' l' l' mlm'l' m'l' ml mlm'l' m'l' ml mlm'l' m'l' ml mlm'l' m'l' ml m where where ⋅ denotes the angular average operator.Utilizing Eq. (R5), the Wigner Dfunctions (see Ref. [53]) and the identity, .
With Eqs.(R9), we can reduce the summations in (R8) and recast it into a more symmetric form: ( ) Then the key step is to express aml and bml in terms of the incident fields.This can be done by taking advantage of the orthogonality among vector spherical harmonic functions (VSHFs) (page 377, Ref. [53]): where and the VSHFs are given by ( , )

C e e
Plugging Eq. ( 5) in the manuscript into (R11) yields the formulas: It follows from Eqs. (R10) and (R13) that Tz ncons is proportional to the momentum curl (∇×p)z for higher multipoles (N > 1).However, Tz ncons should be zero on the dipoles (N = 1), for which l = l′ = 1 and m = 0, 1 in Eq. (R10).This is because the matrix elementsrelated terms in Eq. ( R10) are zero at l = l′ = 1 and m = 0; and for m = 1 (odd) the expansion coefficients-related terms will vanish according to Eq. (R13).On the other hand, one may also work out the expansion coefficients-related terms for the evanescent wave (i.e., imaginary kz = iq): Therefore, the nonconservative torque Tz ncons on the higher multipoles is induced by the reactive helicity gradient (∇H)z for the evanescent wave.Likewise, this gradient torque should also be zero for the dipoles.
The above results regarding the anisotropic particles are incorporated into the manuscript (please see new section "LOT on anisotropic particles" and Fig. 3) and Supplementary Information (Section B).We also add a Method section to elaborate the physics of the conservative and nonconservative torques, from the viewpoint of Fourier series expansion.
Finally, the discussion of the torque in the optical spanner appears disjointed from the remainder of the manuscript.I would suggest to omit the "optical spanner" section, so that the manuscript is more focused.R2.4:We thank the referee for his/her suggestion, but the "optical spanner" section indeed fits well with the principal line of the manuscript.Our work focuses on both the gradient and curl components of the optical torque, and we demonstrate in this section that the curl component is of relevance in the optical spanner configuration.Particularly, this curl component is utilized to realize the negative optical torque (NOT), which is considered "especially important" by the other reviewer.Although the NOT was realized by taking advantage of anisotropic or inhomogeneous materials (cf.Ref. [32][33][34][35][36][37][38][39]), we demonstrate here for the first time that this phenomenon is also possible for the homogeneous isotropic sphere.For these reasons, we would reserve this section in the revised manuscript.
Overall, I miss the broader relevance and the groundbreaking character that would make the manuscript also interesting to experimentalists.Such broader relevance could for instance be expanding the framework to include non-spherical particles.In addition, I am not convinced LOT is interesting/relevant.I therefore cannot recommend publication of the manuscript.It may be better suited to a more specialized journal.

R2.3:
We would mention that spherical particles are of interest to experimentalists.For example, in a recent paper [Nature Nanotechnology 2020, 15, 89-93], the authors employed both dimers and nanospheres for torque sensing, and they chose the nanospheres to explore the possibility of observing the long-sought-after vacuum friction.The spinning microspheres were also used for observing Magnus effects at small scales [Nature Physics, 2023, 19(12): 1904-1909].It is true that experimentally available particles cannot be perfect spheres, but we have demonstrated numerically that our theory works well for rough spheres.
We have also considered a highly anisotropic geometry (the dimer) and demonstrated that the nonconservative component of torque may overcome the restoring one, spinning the dimer continuously (Figs.R1e and f).We develop a framework (i.e., Eq. (R10)) to evaluate the nonconservative torque, which is shown to be induced by the momentum curl (for the travelling wave, Eq. ( R13)) or reactive helicity gradient (for the evanescent wave, Eq. ( R14)).Although we do not present indepth analysis of the restoring or conservative torque, which is beyond the scope of this  2.It would be interesting to include the discussion of the angular momentum conservation in the context of LOT.
Report on the resubmitted paper 481537_1 "Gradient and curl optical torques" by Xiaohao Xu et al.
Upon resubmission, the authors have carefully addressed the previous remarks, and now the paper is essentially improved.It is only one minor point relating the momentum-curl torque (3 rd Eq. ( 3)), that I overlooked in the 1 st review.Actually, in some cases the term p, p being the Poynting vector or a certain its constituent, represents a sort of the "true" electromagnetic spin (see, for example, [47], Transverse and the hidden vorticity of propagating light fields https://doi.org/10.1364/JOSAA.466360;Transverse spinning of unpolarized light https://doi.org/10.1038/s41566-020-00733-3). In such cases, the "momentum-curl" contribution is, at least partially, "captured" by the "pure-spin" torque (1).Accordingly, the "momentum-curl" torque can be "felt" even by dipole particles.Am I right?If so, this point should be briefly commented to avoid confusing ambiguities.

Fig
Fig. R1.FDTD simulation results for different shaped particles.a Sectional view of particles used in the simulations.Particle i represents an ideal sphere with radius of 0.1 µm; particles ii-iv are rough spheres modeled by varying radius values, which converge in distribution to a Gaussian variable with mean 0.1 µm and standard deviation σ = 1, 2 and 3 nm, respectively; particles v and vi are dimers composed of dual spheres with radius of 0.1 and 0.05 µm, respectively.b LOT as a function of the orientation angle αof the spherical particles i-iv.The angle is defined with respect to x-axis, as illustrated by the inset.c Calculated angular potential for the particles i-iv.d LOT and its conservative and nonconservative components versus the orientation angle of the dimer v. e A similar plot for the dimer vi.f Angular potential for v and vi.In all simulations, the particles are illuminated by the travelling wave (case I) in Fig.2a, and are placed at a fixed position x = 0.35λ, where the momentum curl of illumination reaches the maximum.
definition, does no work on the particle rotating a full cycle: particle cannot be rotated continuously under the action of only this torque.In this sense, cons ( ) z T α is responsible for the orientational restoring (or the locked state).
. (7) in the manuscript, we are led to the following results for the traveling wave (i.e., real kz): contribution is reflected by the term related to the scattering cross-section(1)   sca-e C .Consequently, Eq. (R16) is the electric part of our Eq.(1).

P. 11 :
Ref. 62 is incomplete.I think, the suggested corrections are technical and can be made during the final text polishing.I support the paper publication REVIEWERS' COMMENTS Reviewer #1 (Remarks to the Author): I support the paper publication.See the attachment please.[attachment below] ,